A Survey on classifying spaces for families

A <strong>Survey</strong> on classifyingspaces for familiesWolfgang Lück ∗Fachbereich Mathematik und InformatikWestfälische Wilhelms-UniversitätMünsterEinsteinstr. 6248149 MünsterGermanylueck@math.uni-muenster.dehttp://www.math.unimuenster.de/u/lueckJune 2004

1. The G-CW -versionGroup means always locally compact Hausdorfftopological group with a countablebase for its topology.Definition 1. (G-CW -complex) A G-CW -complex X is a G-space together with aG-invariant filtration∅ = X −1 ⊆ X 0 ⊆ . . . ⊆ X n ⊆ . . . ⊆ ⋃n≥0X n = Xsuch that X carries the colimit topologywith respect to this filtration (i.e. a set C ⊆X is closed if and only if C ∩ X n is closedin X n for all n ≥ 0) and X n is obtainedfrom X n−1 for each n ≥ 0 by attachingequivariant n-dimensional cells, i.e. thereexists a G-pushout∐i∈I nG/H i × S n−1⏐↓∐i∈I nG/H i × D n∐i∈In qn i−−−−−−→−−−−−−→ ∐i∈In Qn iX n−1⏐↓X n

Remark 9. (Proper G-spaces) A COMnumerableG-space X is proper. Not everyproper G-space is COM-numerable. But aG-CW -complex X is proper if and only ifit is COM-numerable.Theorem 10. (Homotopy characterizationof J F (G)) Let F be a family of subgroups.1. For any family F there exists a modelfor J F (G) whose isotropy groups belongto F;2. Two models for J F (G) are G-homotopyequivalent;3. For H ∈ F the H-fixed point set J F (G) His contractible.

3. Comparision of the twoversionsThere is always a G-mapφ F : E F (G) → J F (G)which is unique up to G-homotopy.Example 11. Let G be totally disconnected.Thenφ TR : EG → JGis a G-homotopy equivalence if and only ifG is discrete.Theorem 12. (EG = JG)) The G-map φ Fis a G-homotopy equivalence if F = COM,i.e. we get a G-homotopy equivalenceφ COM : EG ≃ −→ JG.Lemma 13. Let G be a totally disconnectedgroup Then the following squarecommutes up to G-homotopy and consistsof G-homotopy equivalencesE COMOP (G) −→ J COMOP (G)⏐↓⏐↓EG −→ JG

4. Special modelsThere are interesting special models• Operator theoretic models;• G/K for an almost connected groupG with K ⊆ G maximal compact subgroup;• Actions on CAT(0)-spaces;• Actions on affine buildings;• The Rips complex for word-hyperbolicgroups;• The Borel-Serre compactification andarithmetic groups;

• Mapping class groups and Teichmüllerspace;• Out(F n ) and outer space.

4.1. Operator Theoretic ModelLet C 0 (G) be the Banach space of complexvalued functions of G vanishing at infinitywith the supremum-norm. The group Gacts isometrically on C 0 (G) by (g ·f)(x) :=f(g −1 x) for f ∈ C 0 (G) and g, x ∈ G. LetP C 0 (G) be the subspace of C 0 (G) consistingof functions f such that f is notidentically zero and has non-negative realnumbers as values.Theorem 14. (Operator theoretic model)The G-space P C 0 (G) is a model for JG.Example 15. Let G be discrete. Anothermodel for JG is the spaceX G = {f : G → [0, 1] | f has finite support,∑g∈Gf(g) = 1}with the topology coming from the supremumnorm.

Remark 16. (Simplicial Model) Let Gbe discrete. Let P ∞ (G) be the geometricrealization of the simplicial set whose k-simplices consist of (k+1)-tupels (g 0 , g 1 , . . . , g k )of elements g i in G. This also a model forEG.The spaces X G and P ∞ (G) have the sameunderlying sets but in general they havedifferent topologies. The identity map inducesa (continuous) G-map P ∞ (G) → X Gwhich is a G-homotopy equivalence, but ingeneral not a G-homeomorphism

4.2. Almost Connected GroupsThe following result is due to Abels.Theorem 17. Almost connected groups)Let G be a (locally compact Hausdorff)topological group. Suppose that G is almostconnected, i.e. the group G/G 0 iscompact for G 0 the component of the identityelement. Then G contains a maximalcompact subgroup K which is unique up toconjugation. The G-space G/K is a modelfor JG.Theorem 18. (Discrete subgroups of almostconnected Lie groups) Let L be aLie group with finitely many path components.Then L contains a maximal compactsubgroup K which is unique up toconjugation. The L-space L/K is a modelfor EL.If G ⊆ L is a discrete subgroup of L, thenL/K with the obvious left G-action is afinite dimensional G-CW -model for EG.

4.3. Actions on CAT(0)-spacesTheorem 19. (Actions on CAT(0)-spaces)Let G be a (locally compact Hausdorff)topological group. Let X be a proper G-CW -complex. Suppose that X has thestructure of a complete CAT(0)-space forwhich G acts by isometries. Then X is amodel for EG.Remark 20. This result contains as specialcase isometric G actions on simplyconnectedcomplete Riemannian manifoldswith non-positive sectional curvature andG-actions on trees.

4.4. Affine BuildingsTheorem 21. (Affine buildings)Let Gbe a totally disconnected group. Supposethat G acts on the affine building by simplicialautomorphisms such that each isotropygroup is compact. Then Σ is a model forboth J COMOP (G) and JG and the barycentricsubdivision Σ ′ is a model for both E COMOP (G)and EG.Example 22 (Bruhat-Tits building). Animportant example is the case of a reductivep-adic algebraic group G and its associatedaffine Bruhat-Tits building β(G).Then β(G) is a model for JG and β(G) ′ isa model for EG by Theorem 21.

4.5. The Rips Complex of aWord-Hyperbolic GroupThe Rips complex P d (G, S) of a group Gwith a symmetric finite set S of generatorsfor a natural number d is the geometricrealization of the simplicial set whoseset of k-simplices consists of (k +1)-tuples(g 0 , g 1 , . . . g k ) of pairwise distinct elementsg i ∈ G satisfying d S (g i , g j ) ≤ d for all i, j ∈{0, 1, . . . , k}. The obvious G-action by simplicialautomorphisms on P d (G, S) inducesa G-action by simplicial automorphisms onthe barycentric subdivision P d (G, S) ′Theorem 23. (Rips complex) Let G bea (discrete) group with a finite symmetricset of generators. Suppose that (G, S) isδ-hyperbolic for the real number δ ≥ 0.Let d be a natural number with d ≥ 16δ +8. Then the barycentric subdivision of theRips complex P d (G, S) ′ is a finite G-CW -model for EG.

4.6. Arithmetic GroupsArithmetic groups in a semisimple connectedlinear Q-algebraic group possess finite modelsfor EG. Namely, let G(R) be the R-points of a semisimple Q-group G(Q) andlet K ⊆ G(R) a maximal compact subgroup.If A ⊆ G(Q) is an arithmetic group,then G(R)/K with the left A-action is amodel for E FIN (A) as already explained inTheorem 18. The A-space G(R)/K is notnecessarily cocompact.Theorem 24. Borel-Serre compactification)The Borel-Serre completion of G(R)/Kis a finite A-CW -model for E FIN (A).

4.7. Mapping Class groupsLet Γ s g,r be the mapping class group ofan orientable compact surface F of genusg with s punctures and r boundary components.We will always assume that 2g +s + r > 2, or, equivalently, that the Eulercharacteristic of the punctured surface Fis negative. It is well-known that the associatedTeichmüller space T s g,r is a contractiblespace on which Γ s g,r acts properly.ActuallyTheorem 25. (Teichmüller space) TheΓ s g,r -space T s g,r is a model for E FIN (Γ s g,r ).

4.8. Outer AutomorphismGroups of Free groupsLet F n be the free group of rank n. Denoteby Out(F n ) the group of outer automorphismsof F n , i.e. the quotient of the groupof all automorphisms of F n by the normalsubgroup of inner automorphisms. Cullerand Vogtmann have constructed a spaceX n called outer space on which Out(F n )acts with finite isotropy groups. It is analogousto the Teichmüller space of a surfacewith the action of the mapping classgroup of the surface.The space X n contains a spine K n whichis an Out(F n )-equivariant deformation retraction.This space K n is a simplicialcomplex of dimension (2n − 3) on whichthe Out(F n )-action is by simplicial automorphismsand cocompact. Actually thegroup of simplicial automorphisms of K nis Out(F n ) by results due to Bridson andVogtman.Theorem 26. The barycentric subdivisionK n ′ is a finite (2n − 3)-dimensional modelof E Out(F n ).

5. Relevance and Applicationsof Classifying Spaces forFamilies5.1. Baum-Connes ConjectureThe goal of the Baum-Connes Conjectureis the computation of the topological K-theory K n (C ∗ r (G)) of the reduced groupC ∗ -algebra of G.Conjecture 27 (Baum-Connes Conjecture).The assembly map defined by takingthe equivariant indexasmb: K G n (JG) ∼ = −→ Kn (C ∗ r (G))is bijective for all n ∈ Z.

5.2. Farrell-Jones ConjectureLet G be a discrete group.associative ring with unit.Let R be aThe goal ofthe Farrell-Jones Conjecture is to computethe algebraic K-groups K n (RH) and thealgebraic L-groups L −∞ n (RG).Conjecture 28 (Farrell-Jones Conjecture).The assembly maps induced by theprojection E VCYC (G) → G/Gasmb: Hn G (E VCYC (G), K) → K n (RG);asmb: Hn G (E VCYC (G), L −∞ ) → L −∞ n (RG),are bijective for all n ∈ Z.

5.3. Completion TheoremLet G be a discrete group. For a proper finiteG-CW -complex let KG ∗ (X) be its equivariantK-theory defined in terms of equivariantfinite dimensional complex vector bundlesover X. Let I ⊆ KG 0 (EG) be theaugmentation ideal, i.e. the kernel of themap K 0 (EG) → Z sending the class ofan equivariant complex vector bundle toits complex dimension. Let KG ∗ (EG)Î bethe I-adic completion of KG ∗ (EG) and letK ∗ (BG) be the topological K-theory ofBG.Theorem 29 (Completion Theorem fordiscrete groups). Let G be a discrete groupsuch that there exists a finite model forEG. Then there is a canonical isomorphismK ∗ (BG) ∼ = −→ K∗G (EG) Î .

5.4. Classifying Spaces forEquivariant BundlesThe equivariant K-theory for finite properG-CW -complexes appearing above can beextended to arbitrary proper G-CW -complexes(including the multiplicative structure) usingΓ-spaces in the sense of Segal and involvingclassifying spaces for equivariantvector bundles. These classifying spacesfor equivariant vector bundles are againclassifying spaces of certain Lie groups andcertain families5.5. Equivariant Homology andCohomologyClassifying spaces for families play a role incomputations of equivariant homology andcohomology for compact Lie groups suchas equivariant bordism. Rational computationsof equivariant (co-)-homology groupsare possible in general using Chern charactersfor discrete groups and proper G-CW -complexes

6. Finiteness ConditionsThe questions whether there exists finitemodels, models of finite type or models orfinite-dimensional models for EG or wthatis the minimal value of dim(EG) is quiteinteresting and an obvious extention of thesame question for BG.Remark 30 (Algebraic criterion). Let Gbe discrete. In the classical case one canread off the possible dimension of BG fromthe homological algebra of ZG, in particularin terms of the cohomological dimensionof the trivial ZG-modul Z. There areanalogous results for E F (G) if one considersmodules over the orbit category Or(G),in particular the constant contravariant ZOr(G)-module Z F whose value is Z on G/H forH ∈ F and {0} on G/H for H ∉ F. Thisgives in principle a complete answer in algebraicterms but is often hard to apply inconcrete situations.

6.1. Some conditions forfinite-dimensional modelsAs an illustration we give a small selectionof results on this topic to due to Connolly,Dunwoody, Kropholler, Kozniewsky,L., Leary, Meintrup, Mislin and Nucinkisand others.Theorem 31 (Discrete subgroups of Liegroups). Let L be a Lie group with finitelymany path components. Let K ⊆ L be amaximal compact subgroup K. Let G ⊆ Lbe a discrete subgroup of L.Then L/K with the left G-action is a modelfor EG.Suppose additionally that G contains a torsionfreesubgroup ∆ ⊆ G of finite index.Then we havevcd(G) ≤ dim(L/K)and equality holds if and only if G\L iscompact.

Theorem 32 (A criterion for 1-dimensionalmodels). Let G be a discrete group. Thenthere exists a 1-dimensional model for EGif and only the cohomological dimension ofG over the rationals Q is less or equal toone.Theorem 33. Virtual cohomological dimensionand dim(EG) Let G be a discretegroup which contains a torsionfree subgroupof finite index and has virtual cohomologicaldimensionvcd(G) ≤ d. Let l ≥0 be an integer such that the length l(H)of any finite subgroup H ⊂ G is boundedby l.Then we have vcd(G) ≤ dim(EG) for anymodel for EG and there exists a model forEG of dimension max{3, d} + l.

Example 34 (Virtually poly-cyclic groups).Let the group ∆ be virtually poly-cyclic,i.e. ∆ contains a subgroup ∆ ′ of finite indexfor which there is a finite sequence{1} = ∆ ′ 0 ⊆ ∆′ 1 ⊆ . . . ⊆ ∆′ n = ∆ ′ of subgroupssuch that ∆ ′ i−1 is normal in ∆′ i withcyclic quotient ∆ ′ i /∆′ i−1for i = 1, 2, . . . , n.Denote by r the number of elements i ∈{1, 2, . . . , n} with ∆ ′ i /∆′ ∼ i−1 = Z. The numberr is called the Hirsch rank. The group∆ contains a torsionfree subgroup of finiteindex. We call ∆ ′ poly-Z if r = n, i.e.all quotients ∆ ′ i /∆′ i−1are infinite cyclic.Then1. r = vcd(∆);2. r = max{i | H i (∆ ′ ; Z/2) ≠ 0} for one(and hence all) poly-Z subgroup ∆ ′ ⊂∆ of finite index;3. There exists a finite r-dimensional modelfor E∆ and for any model E∆ we havedim(E∆) ≥ r.

6.2. Reduction to discretegroupsThe discretization G d of a topological groupG is the same group but now with the discretetopology.Theorem 35 (Passage from topologicalgroups to totally disconnected groups).Let G be a locally compact second countableHausdorff group. Put G := G/G 0 .Then there is a G-CW -model for EG thatis d-dimensional or finite or of finite typerespectively if and only if EG has a G-CW -model that is d-dimensional or finite or offinite type respectively.Theorem 36 (Passage from totally disconnectedgroups to discrete groups).Let G be a locally compact totally disconnectedHausdorff group and let F bea family of subgroups of G. Then there is aG-CW -model for E F (G) that is d-dimensionalor finite or of finite type respectively if andonly if there is a G d -CW -model for E F (G d )that is d-dimensional or finite or of finitetype respectively.

7. CounterexamplesThe following problem is stated by BrownIt created a lot of activities and many ofthe results stated above were motivated byit.Problem 37. For which discrete groups G,which contain a torsionfree subgroup offinite index and has virtual cohomologicaldimension ≤ d, does there exist a d-dimensional G-CW -model for EG?Leary and Nucinkis have constructed manyvery interesting examples of discrete groupssome of which are listed below. Theirmain technical input is an equivariant versionof the constructions due to Bestvinaand Brady. These examples show that theanswer to the Problems 37 and to otherproblems appearing in the literature is notpositive in general. A group G is of typeVF if it contains a subgroup H ⊆ G of finiteindex for which there is a finite modelfor BH.

1. For any positive integer d there exist agroup G of type VF which has virtuallycohomological dimension ≤ 3d, but forwhich any model for EG has dimension≥ 4d;2. There exists a group G with a finitecyclic subgroup H ⊆ G such that G isof type VF but the centralizer C G H ofH in G is not of type FP ∞ ;3. There exists a group G of type VFwhich contains infinitely many conjugacyclasses of finite subgroups;4. There exists an extension 1 → ∆ →G → π → 1 such that E∆ and EG havefinite G-CW -models but there is no G-CW -model for Eπ of finite type.

8. The Orbit Space of EGWe will see that in many computations ofthe group (co-)homology, of the algebraicK- and L-theory of the group ring or thetopological K-theory of the reduced C ∗ -algebra of a discrete group G a key problemis to determine the homotopy type ofthe quotient space G\EG of EG. The followingresult shows that this is a difficultproblem in general and can only be solvedin special cases where som extra geometricinput is available. It was proved byLeary and Nucinkis based on ideas due toBaumslag-Dyer-Heller and Kan and Thurston.Theorem 38 (The homotopy type ofG\EG). Let X be a connected CW -complex.Then there exists a group G such thatG\EG is homotopy equivalent to X.